Three tanks have capacities 98 litres, 182 litres, and 266 litres. What is the capacity, in litres, of the largest measuring cylinder that can exactly fill each of the three tanks an integer number of times?

Introduction:
This question applies the concept of the highest common factor (HCF) to a real life situation involving measuring liquids. We want the largest possible measuring cylinder that can measure all three given quantities exactly, without leaving any remainder in any tank.

Given Data / Assumptions:

• Tank capacities: 98 litres, 182 litres, and 266 litres
• We need the largest possible cylinder capacity that will divide each capacity exactly.
• Units of all capacities are in litres.

Concept / Approach:
To find the largest measure that can exactly measure all given quantities, we need the HCF of the three numbers. The HCF is the greatest positive integer that divides each number without leaving a remainder. Prime factorisation or the Euclidean algorithm can be used to find this HCF.

Step-by-Step Solution:
Step 1: Find HCF of 98 and 182: 98 = 2 × 7^2 182 = 2 × 7 × 13 Common primes: 2 × 7 = 14, so HCF(98, 182) = 14 Step 2: Find HCF of the result with 266: 266 = 2 × 7 × 19 HCF(14, 266): 14 = 2 × 7, and 266 has 2 and 7 as factors, so HCF = 14 Therefore, HCF(98, 182, 266) = 14 So the largest measuring cylinder that can be used is 14 litres

Verification / Alternative check:
We can verify by division: 98 ÷ 14 = 7, 182 ÷ 14 = 13, and 266 ÷ 14 = 19. All results are integers, confirming that 14 litres exactly measures each capacity. No larger common divisor exists, because any larger number would not divide at least one of the capacities without remainder.

Why Other Options Are Wrong:
7 litres divides all three capacities, but it is not the largest possible measure. 42 litres does not divide all the capacities exactly (for example, 182 ÷ 42 is not an integer). 98 litres also fails because it does not divide 182 or 266 evenly. Hence only 14 litres is correct.

Common Pitfalls:
Learners may confuse HCF with LCM and attempt to find a common multiple instead of a common divisor. Another mistake is to choose a measure that works for two of the tanks but not for the third. Always confirm that the chosen measure divides every given capacity exactly.

Final Answer:
The capacity of the largest measuring cylinder that can exactly measure all three tanks is 14 litres.